Position change of body in non-isolated system

[meathead]

Homework Statement

A body of mass 5 kg is initially at rest. At time t=0, a constantly increasing force is applied to the body for 7.5 seconds according to:F(t)=at; where a is a constant 0.866 N/s. Determine the distance x the body travels during the application of the force.

Homework Equations

delta E of system = sigma T
sum of forces in y direction =0
normal force - mass*gravitational acceleration=0
work=force*change in position
1/2mass*final velocity^2 = 1/2mass*initial velocity^2 + work

The Attempt at a Solution

At kind of a loss for where to start this problem because the force is not constant.
Can I some how find a value for Work and use that to solve for final velocity and use that to find change in position? Please Help!

 

[ehild]

You can not use the work-energy theorem in this problem.
Use Newton's Law: the acceleration is force/mass. Now the acceleration depends on time. Acceleration is the time derivative of velocity; you get velocity by integrating acceleration:

$$v(t) = \int{a(t)dt}$$

the integration constant is detemined from the condition v(0)=0.
Velocity is the time derivative of displacement: you get the displacement by integrating velocity: v(t)

$$x(t=7.5) -x(0) = \int_0^{t=7.5}{v(t)dt}$$



ehild

 

[kerimek]

I would approach this problem by first analyzing the given information and identifying the relevant equations and principles to use. In this case, we are dealing with a non-isolated system (since an external force is being applied) and we are interested in determining the position change of the body.

To start, we can use the equation for work, W = F*d, where F is the force and d is the distance traveled. Since we know the force, F(t) = at, we can integrate this equation to find the work done over the given time period of 7.5 seconds:

W = ∫ F(t) dt = ∫ at dt = (1/2)at^2

Plugging in the given values, we get W = (1/2)(0.866 N/s)(7.5 s)^2 = 24.56 J

Next, we can use the equation for kinetic energy, KE = (1/2)mv^2, to find the final velocity of the body. Since the body is initially at rest, we can set the initial kinetic energy to zero and solve for the final velocity:

KE = (1/2)mv^2 = W
v = √(2W/m) = √(2*24.56 J/5 kg) = 2.48 m/s

Finally, we can use the equation for position change, Δx = v*t, to find the distance traveled by the body:

Δx = v*t = (2.48 m/s)(7.5 s) = 18.6 m

Therefore, the body travels a distance of 18.6 meters during the application of the force.

In summary, as a scientist, I would approach this problem by identifying the relevant equations and principles to use and then applying them step by step to find the solution. By using the equations for work, kinetic energy, and position change, we were able to determine the distance traveled by the body in this non-isolated system.